Question: Simplify and expand the following expression: $ \dfrac{p}{2p - 6}+\dfrac{5p}{p + 1} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2p - 6)(p + 1)$ Multiply the first term by $\dfrac{p + 1}{p + 1}$ $ \begin{align*} \dfrac{p}{2p - 6} \times \dfrac{p + 1}{p + 1} & = \dfrac{(p)(p + 1)}{(2p - 6)(p + 1)} \\ & = \dfrac{p^2 + p}{(2p - 6)(p + 1)}\end{align*} $ Multiply the second term by $\dfrac{2p - 6}{2p - 6}$ $ \begin{align*} \dfrac{5p}{p + 1} \times \dfrac{2p - 6}{2p - 6} & = \dfrac{(5p)(2p - 6)}{(p + 1)(2p - 6)} \\ & = \dfrac{10p^2 - 30p}{(p + 1)(2p - 6)}\end{align*} $ Now we have: $ = \dfrac{p^2 + p}{(2p - 6)(p + 1)} + \dfrac{10p^2 - 30p}{(p + 1)(2p - 6)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{p^2 + p + 10p^2 - 30p}{(2p - 6)(p + 1)} $ $ = \dfrac{11p^2 - 29p}{(2p - 6)(p + 1)}$ Expand the denominator: $ = \dfrac{11p^2 - 29p}{2p^2 - 4p - 6}$